Binomial Theorem

The remainder when $7^{2022}+3^{2022}$ is divided by 5 is :

Consider the sequence $a_1,a_2,a_3,\ldots$ such that $a_1=1$, $a_2=2$ and $a_{n+2}=\dfrac2{a_{n+1}}+a_n$ for $n=1,2,3,\ldots$. If

$$\left(\frac{a_1+\frac1{a_2}}{a_3}\right)\cdot\left(\frac{a_2+\frac1{a_3}}{a_4}\right)\cdot\left(\frac{a_3+\frac1{a_4}}{a_5}\right)\cdots\left(\frac{a_{30}+\frac1{a_{31}}}{a_{32}}\right)=2^\alpha\left({}^{61}C_{31}\right),$$

then $\alpha$ is equal to :

Let the coefficients of the middle terms in the expansion of $\left(\dfrac1{\sqrt6}+\beta x\right)^4$, $(1-3\beta x)^2$ and $\left(1-\dfrac\beta2x\right)^6$, $\beta>0$, respectively form the first three terms of an A.P. If d is the common difference of this A.P., then $50-\dfrac{2d}{\beta^2}$ is equal to ______.

If $1+\left(2+{}^{49}C_1+{}^{49}C_2+\ldots+{}^{49}C_{49}\right)\left({}^{50}C_2+{}^{50}C_4+\ldots+{}^{50}C_{50}\right)$ is equal to $2^n\cdot m$, where m is odd, then n + m is equal to ______.

If $z=2+3i$, then $z^5+(\bar z)^5$ is equal to :

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $\left(\sqrt[4]2+\dfrac1{\sqrt[4]3}\right)^n$, in the increasing powers of $\dfrac1{\sqrt[4]3}$ be $\sqrt[4]6:1$. If the sixth term from the beginning is $\dfrac\alpha{\sqrt[4]3}$, then α is equal to _______.

If $\displaystyle\sum_{k=1}^{10}K^2\left({}^{10}C_K\right)^2=22000L$, then L is equal to _______.

The coefficient of $x^{1011}$ in the expression $(1+2x)^{2022}+2x(1+2x)^{2021}+(2x)^2(1+2x)^{2020}+\dots+(2x)^{2022}$ is

If $p_1$ and $p_2$, respectively, are the second largest and the fourth largest prime numbers in the factorization of the coefficient of $x^{30}$ in the expression of $\left(x^2+2+\dfrac1{x^2}\right)^{-5}(1+x^2)^{40}$, then $p_1+p_2$ is equal to _______

The value of $-{}^{15}C_1+2\cdot{}^{15}C_2-3\cdot{}^{15}C_3+\dots-15\cdot{}^{15}C_{15}+{}^{14}C_1+{}^{14}C_3+{}^{14}C_5+\dots+{}^{14}C_{11}$ is :

For nonnegative integers $s$ and $r$, let

$$\binom sr = \begin{cases}\dfrac{s!}{r!\,(s-r)!} & \text{if } r \le s,\\ 0 & \text{if } r > s.\end{cases}$$

For positive integers $m$ and $n$, let

$$g(m,n) = \sum_{p=0}^{m+n}\frac{f(m,n,p)}{\binom{n+p}{p}}$$

where for any nonnegative integer $p$,

$$f(m,n,p) = \sum_{i=0}^{p}\binom mi\binom{n+i}{p}\binom{p+n}{p-i}.$$

Then which of the following statements is/are TRUE?

Suppose

$$\det\begin{bmatrix}\displaystyle\sum_{k=0}^nk & \displaystyle\sum_{k=0}^n{}^nC_k\,k^2\\ \displaystyle\sum_{k=0}^n{}^nC_k\,k & \displaystyle\sum_{k=0}^n{}^nC_k\,3^k\end{bmatrix} = 0$$

holds for some positive integer $n$. Then $\displaystyle\sum_{k=0}^n\frac{{}^nC_k}{k+1}$ equals ______

Let

$$X = \left({}^{10}C_1\right)^2 + 2\left({}^{10}C_2\right)^2 + 3\left({}^{10}C_3\right)^2 + \cdots + 10\left({}^{10}C_{10}\right)^2,$$

where ${}^{10}C_r$, $r \in \{1, 2, \cdots, 10\}$ denote binomial coefficients. Then, the value of $\frac{1}{1430}X$ is _____ .

In a high school, a committee has to be formed from a group of 6 boys $M_1, M_2, M_3, M_4, M_5, M_6$ and 5 girls $G_1, G_2, G_3, G_4, G_5$.

(i) Let $\alpha_1$ be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.

(ii) Let $\alpha_2$ be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.

(iii) Let $\alpha_3$ be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.

(iv) Let $\alpha_4$ be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls and such that both $M_1$ and $G_1$ are NOT in the committee together.

The correct option is:

The value of
$$\left({}^{21}C_1-{}^{10}C_1\right)+\left({}^{21}C_2-{}^{10}C_2\right)+\left({}^{21}C_3-{}^{10}C_3\right)+\left({}^{21}C_4-{}^{10}C_4\right)+\dots+\left({}^{21}C_{10}-{}^{10}C_{10}\right)$$
is :

If $(27)^{999}$ is divided by 7, then the remainder is :

The coefficient of $x^{-5}$ in the binomial expansion of $\left(\dfrac{x+1}{x^{2/3}-x^{1/3}+1}-\dfrac{x-1}{x-x^{1/2}}\right)^{10}$, where $x\ne0,1$, is :

Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x)^2+(1+x)^3+\cdots+(1+x)^{49}+(1+mx)^{50}$ is $(3n+1)\,{}^{51}C_3$ for some positive integer $n$. Then the value of $n$ is

If the number of terms in the expansion of $\left(1-\dfrac2x+\dfrac4{x^2}\right)^n$, $x\ne0$, is 28, then the sum of the coefficients of all the terms in this expansion, is :

For $x\in\mathbb R$, $x\ne-1$, if $(1+x)^{2016}+x(1+x)^{2015}+x^2(1+x)^{2014}+\dots+x^{2016}=\displaystyle\sum_{i=0}^{2016}a_ix^i$, then $a_{17}$ is equal to :